Mathematics

LIKE TO JOIN ON FACEBOOK ( click here ) Ques: Show that all the roots of the equation   where   ; i = 1,2,3,4 lie outside the circle |z| = 2/3 ? Solution: For proving the above problem, first we assume the contrary of the same that roots of the equation lie inside or 'on' the circle i.e.    ,   Now, Taking mod of both the sides, (for properties of modulus of complex number [ click here ] ) or,     ( since   ) [ since  ; i = 1,2,3,4 , therefore    ] or,  or,    ( by sum of Geometric Progression ) Now,    as,  but we assume that    ( Less than or equal to 1.4 ) ............ (i) but   ( greater than or equal to 2 ) ..............(ii) from (i) and (ii), it is a contradiction. Hence   is not true. Therefore |z| = 2/3 and hence roots are lie outside the circle |z| = 2/3 ( Hence Proved ) Ques: Find the l...

LIKE TO JOIN ON FACEBOOK ( click here ) Ques 1: Find the number of solutions of the equations,    and |z - 3 + i| = 3 ? Sol: 1 . Equation   is an equation of arc of the circle . The angle should be measured from denominator z - 4 - 3i to numerator z - 2 - i. End points of the arc (chord) of the circle are (4, 3) and (2, 1) distance between (4,3) and (2,1) = 2 root 2 Angle at centre = pi/2 so, radius = 2 Centre: (2,2) 2. Equation |z - 3 + i| = 3 is an equation of circle. Centre (3, -1) and radius = 3 Draw the above graph of circle, find the position of (2,1) in respect of the circle, |2 + i - 3 + i| - 3 = root 5 - 3 < 0 therefore point (2,1) lies inside the circle   |z - 3 + i| = 3 Therefore, from the figure it is clear that both the graphs cut each other only on one point. Therefore, there is only one common point on both the curve. So, there is only one solution exist for the given equations...